Compute the 9th derivative

s3a

Nov 2008
624
5
How do I do this the way I am supposed to? (without differentiating 9 times)

Any help would be greatly appreciated!
Thanks in advance!
 

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Jester

MHF Helper
Dec 2008
2,470
1,255
Conway AR
How do I do this the way I am supposed to? (without differentiating 9 times)

Any help would be greatly appreciated!
Thanks in advance!
Use the hint. The Taylor series for \(\displaystyle \tan^{-1}x\) is

\(\displaystyle
\tan^{-1}x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots.
\)

Subs in for \(\displaystyle x \rightarrow \frac{x^3}{4}\). You really only need to consider powers up to \(\displaystyle x^9\).
 
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skeeter

MHF Helper
Jun 2008
16,216
6,764
North Texas
How do I do this the way I am supposed to? (without differentiating 9 times)

Any help would be greatly appreciated!
Thanks in advance!
did you use the hint?

\(\displaystyle f(x) = \arctan\left(\frac{x^3}{4}\right) = \frac{x^3}{4} - \frac{x^9}{4^3 \cdot 3} + \frac{x^{15}}{4^5 \cdot 5} - \frac{x^{21}}{4^7 \cdot 7} + ...\)

\(\displaystyle f^{(9)}(0) = -\frac{9!}{4^3 \cdot 3}\)
 
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s3a

Nov 2008
624
5
For arctan(x) I get = sigma from 0 to inf of (-x^(2n)) which gives - x - x^3 /3 - x^5 /5 etc without the alternating. What did I do wrong so far? I did d/dx arctan(x) = 1/(1-(-x^2)) then integrate the sum.
 

skeeter

MHF Helper
Jun 2008
16,216
6,764
North Texas
For arctan(x) I get = sigma from 0 to inf of (-x^(2n)) which gives - x - x^3 /3 - x^5 /5 etc without the alternating. What did I do wrong so far? I did d/dx arctan(x) = 1/(1-(-x^2)) then integrate the sum.
\(\displaystyle \frac{1}{1-r} = 1 + r + r^2 + r^3 + ...\)

\(\displaystyle \frac{1}{1-(-x^2)} = 1 + (-x^2) + (-x^2)^2 + (-x^2)^3 + ... = 1 - x^2 + x^4 - x^6 + ...\)
 
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s3a

Nov 2008
624
5
Ok I see how to get the summation of arctan(x^3 /4) but then I don't know what to do. Do I just write down the polynomials up to the x^9 power and do something there? What about f^(n) (a) x^n /n! formula ? What I was trying to do before my current attempt was to just do the summation of f^(n) (a) x^n /n! and to replace f^(n) with it's appropriate algebraic expression but it was tough to compute the second derivative and on.

Sorry for being stupid :(
 
Jan 2010
354
173
So you're OK with this, correct?

\(\displaystyle f(x) = \arctan \left(\frac{x^3}{4}\right) = \frac{x^3}{4} - \frac{x^9}{4^3 \cdot 3} + \frac{x^{15}}{4^5 \cdot 5} - \frac{x^{21}}{4^7 \cdot 7} + \cdots\)

Now, consider what happens when you take 9 derivatives of this expression.

The first term \(\displaystyle \frac{x^3}{4}\) will eventually become zero (beyond 3 derivatives).

The second term \(\displaystyle -\frac{x^9}{4^3 \cdot 3}\) will become a constant. What constant does it become?

The remaining terms will turn into the form \(\displaystyle C \cdot x^n\) where \(\displaystyle C\) is a constant and \(\displaystyle n\) is a positive integer. When you substitute \(\displaystyle x=0\) into a term of this form, does it make sense that these terms will all become zero?

Therefore, there is only one nonzero term, which is the second term.
 
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s3a

Nov 2008
624
5
OMG! Yes, I get it now, thanks!
 
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